Let's say you want to calculate a VaR for a portfolio of 1000 stocks. You're really only interested in the left tail, so do you use the whole set of returns to estimate mean, variance, skew, and shape (let's also assume a skewed generalized error distribution - SGED)? Or would you just use the left tail (let's say the bottom 10% of returns)?

For some reason, using the whole set of returns seems more correct to me (by using only the left 10% or returns you'd really be approaching a non-parametric VaR). But using the whole set of returns would likely cause some distortion in the left tail in order to get a better fit elsewhere.

2 Answers
2

Perhaps you may want to consider article by D. Levine - Modeling Tail Behavior with Extreme Value Theory who gives practicale example on how EVT can be used to calculate probabilities on returns in tails with use of the Pickands-Balkema-de Haan Theorem and generalized Pareto distribution. It also contains some criterias and points on other methods that can be used to determine threshold value for PBH theorem:

Contrary to this notion is the fact that the PBH theorem
states a result based on the assumption of threshold values
approaching the right endpoint of the distribution F. This
implies that better GPD fits are expected for larger choices
of the threshold u.

One must strike a balance between choosing u large
enough so that the theorem is applicable from a practical
standpoint and small enough so that a sufficient number
of data points can be used in estimation of the parameters
of the GPD.

Good find! I overlooked actuaries. Levine provides a nice discussion on how he suggests estimating the distribution's parameters. The motivation for my question where some papers using ES and VaR that said "use the left 10%" and it seemed like it shouldn't be a blanket statement. Thanks!
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Richard HerronFeb 9 '11 at 3:04

One approach is Conditional Value at Risk (CVaR) a.k.a. Expected Shortfall (ES). It does, as you suggest, take into account the whole set of returns. However, instead of traditional VaR which asks "what is the worst 1% or 5% loss I can expect" in a given time frame, conditional VaR asks "assuming I sustain losses of at least 95% or 99% (and perhaps am capitalized to sustain losses of only this amount), what is my expected loss (or shortfall)" for this time period? It can be argued this is more relevant for understanding the impact of more dire scenarios.

Another approach from Extreme Value Theory is concerned with strictly modeling the heavy tail of the returns. Generalized distributions (e.g. Gumbel, Frechet) can be fit to the tail(s) in question via something called the Hill Estimation technique. These are covered in depth in literature should you be interested in more detail.

Thanks! I am familiar with techniques and calculations given the return distribution. But how do people tend to estimate this distribution? It seems the distribution parameters would be very different if we used all returns instead of the left 10%, or the left 5% in the curve fitting. Do you have any idea what portion of the return observations most use in practice?
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Richard HerronFeb 8 '11 at 20:27